- Identify the side lengths of the triangle as \(a=10\), \(b=12\), and \(c=15\).
- Calculate the squares of the side lengths: \(a^2=100\), \(b^2=144\), and \(c^2=225\).
- Compare the sum of the squares of the two shorter sides, \(a^2+b^2=244\), with the square of the longest side, \(c^2=225\).
- Since \(a^2+b^2 > c^2\), the triangle is acute, making option (c) the correct classification: \(\boxed{c}\).
Explanation
- Classifying the triangle based on side lengths
To classify a triangle with side lengths \(a=10\), \(b=12\), and \(c=15\), we use the relationship between the sum of the squares of the two shorter sides and the square of the longest side.
First, identify the sides:
The lengths are 10 in., 12 in., and 15 in.
The longest side is \(c = 15\). The other two sides are \(a = 10\) and \(b = 12\).
Next, calculate the squares of these lengths:
\(a^2 = 10^2 = 100\)
\(b^2 = 12^2 = 144\)
\(c^2 = 15^2 = 225\)
Now, compare the sum of the squares of the two shorter sides to the square of the longest side:
\(a^2 + b^2 = 100 + 144 = 244\)
We compare this result to \(c^2 = 225\).
Since \(244 > 225\), which means \(a^2 + b^2 > c^2\), the triangle is acute.
Finally, evaluate the given options:
Aobtuse, because \(12^2 + 15^2 > 10^2\) (Incorrect reasoning and conclusion)
Bacute, because \(12^2 + 15^2 > 10^2\) (Correct conclusion, but incorrect reasoning/logic)
Cacute, because \(10^2 + 12^2 > 15^2\) (Correct conclusion and correct reasoning)
Thus, option (c) is the correct choice.
Examples
This classification method is essential in architecture and construction, such as when framing a roof or building a support structure. By knowing if a triangle is acute, right, or obtuse, you can determine if the corners are sharp, square, or wide, which is crucial for structural stability and fitting components together precisely. For instance, knowing a triangle is acute helps in ensuring that roof trusses distribute weight effectively across the supporting walls.